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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"c942704ac207d77e07f1d37cc98c10e2be95efc0dc3e923bc273c222c8d9a38e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e03395d80bc5749d","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           1;\n  1:          15,           4;\n  2:         209,         120,          16;\n  3:        2783,        2508,         720,          64;\n  4:       35937,       44528,       20064,        3840,         256;\n  5:      453871,      718740,      445280,      133760,       19200,       1024;\n  6:     5636785,    10892904,     8624880,     3562240,      802560,      92160,      4096;\n  7:    69090879,   157829980,   152500656,    80498880,    24935680,    4494336,    430080,     16384;\n  8:   837948353,  2210908128,  2525279680,  1626673664,   643991040,  159588352,  23969792,   1966080,   65536;\n  9: 10074867407, 30166140708, 39796346304, 30303356160, 14640062976, 4636735488, 957530112, 123273216, 8847360, 262144;\n","recurrence_sentence":"The row polynomials satisfy (4*n+7+4*x)*P(n,x) = (44*n+121+(16*n+88)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((44*n + 121 + (16*n + 88)*x + 16*x^2)*P[n - 1, x])/(4*n + 7 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((44*n + 121 + (16*n + 88)*x + 16*x^2)*v[i - 1])/(4*n + 7 + 4*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013611","D+1"],"edges":["e-551d764375152d3c","e-4578e8ba3c4f1e64"],"text":"A038303 \u2192 basis A013611 \u2192 D+1"},"headline":"A038303 \u2192 basis A013611 \u2192 D+1","chain":null}